Let \(P^\star \)  denote the true distribution of \(Y_1\)  or of \((X_1,Y_1)\) . If the model is correctly specified, then there exists a \(\theta ^\star \in \Theta \)  such that \(P^\star = P_{\theta ^\star }\) . In that case, under suitable regularity conditions, inference based on the posterior distribution will be valid, at least asymptotically. That is, \(\Pi _n\)  will concentrate its mass around \(\theta \)  as \(n \rightarrow \infty \)  and, moreover, the Bernstein–von Mises theorem [1]}, states that \(\Pi _n\)  is approximately a normal distribution, centered at the maximum likelihood estimator \(\hat{\theta }_n\) , with covariance matrix proportional to the inverse of the Fisher information matrix at \(\theta ^\star \) . This implies, among other things, that credible regions derived from \(\Pi _n\)  closely resemble those asymptotic confidence regions based on likelihood theory. Therefore, asymptotically, the Bayesian posterior credible regions will have frequentist coverage probability close to the advertised level.
